Had a final interview, and my internship asked me to come on full-time, do I tell the recruiter?

Let me know if more background is needed.

I've had five excellent interviews for an administrative assistant position at a high-profile, prominent multinational company. I am currently an intern for my state's government, and they have (verbally) asked me to stay on indefinitely, and, should I wish, transition into a full-time position with a 50% raise, benefits, and so on.

The (internal) recruiter I've been working with has been very professional in informing me of delays (such a welcome break from ghosting), and was aware from the get-go that any one of my past internships would convert to a full-time offer. Currently, I'm not set to hear back until a week from now, and would hypothetically start full-time at my current internship in two weeks.

Should I inform the recruiter now that I've received a verbal offer, or wait until it's all on paper (which may not be for a couple of weeks due to the state of, well, the world)? Or, should I never tell them at all? These jobs have incredibly different demands and functions.

TL;DR - internship made verbal FT offer, do I let the recruiter for my 'most ideal' job know?

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